Hard exclusive processes have attracted much attention in recent years. The main interest has been
focussed on the deeply virtual processes for which the virtuality of the photon provides
the hard scale. Examples for such processes are deep-virtual Compton scattering or meson
production (DVMP). The basis of the theoretical treatment of these processes is the representation
of the amplitudes as a convolution of a hard partonic subprocess and soft hadronic matrix
elements, parameterized as GPDs [1]}, [2]}. Besides the deeply virtual processes
also wide-angle ones have been investigated for which the hard scale is provided by a large
momentum transfer from the photon to the meson. Large momentum transfer is equivalent to large
Mandelstam variables \(s, -t\)  and \(-u\) .
The first process of this type, studied within this handbag mechanism, has been wide-angle Compton
scattering (WACS). There are reasonable arguments [3]}, [4]} that, for this kinematical situation,
the Compton amplitude factorize into a product of perturbatively calculable partonic subprocess amplitudes
and form factors of the nucleon representing \(1/x\) -moments of zero-skewness GPDs. Since the relevant GPDs
in this case, \(H\) , \(E\)  and \(\widetilde{H}\)  for valence quarks are known from an analysis of the nucleon's
electromagnetic form factors [5]}, [6]} one can compute the Compton form factors and subsequently
the Compton cross section. The results of this parameter-free prediction [6]} agree fairly well with
experiment [8]}.
The authors of Ref. [9]} extended the approach advocated for in [4]}
to photoproduction of uncharged pions and \(\rho \)  mesons.
It however turned out that the calculated
cross sections are way below the experimental data. An attempt [11]} to
improve this (twist-2) analysis by taking into account twist-3 contributions in the Wandzura-Wilczek
(WW) approximation,
i.e. with only the \(q\bar{q}\)  Fock component of the pion considered, failed too - the
twist-3 contribution in WW approximation is zero in wide-angle photoproduction of pions. This is in
sharp contrast to DVMP where the WW approximation is non-zero [12]}, [13]} and plays an important
role in the interpretation of the data on deeply virtual pion electroproduction [14]}, [15]}, [16]}.
With the arrival of the CLAS data [17]} on wide-angle photoproduction of \(\pi ^0\)  mesons the
handbag mechanism has again been taken up by us [18]}. Now, as a further development of the previous
study [11]}, the full twist-3 contribution to the subprocess amplitude is included in the analysis.
Both its parts, the one from the \(q\bar{q}\)  Fock component of the pion as well as that from its
\(q\bar{q}g\)  Fock component, are connected by the equations of motion and both are required to accomplish
gauge invariance. The twist-3 contributions to the hard subprocess go along with leading-twist
transversity GPDs, \(H_T\) , \(\bar{E}_T\)  and \(\widetilde{H}_T\)  and the corresponding form factors. We stress
that this twist-3 effect, although being formally suppressed by \(\mu _\pi /\sqrt{s}\)  at the amplitude level,
is very strong due to the large mass parameter, \(\mu _\pi \) . As shown
in [18]} the twist-3 contribution dominates the twist-2 one analogously to
the case of deeply virtual \(\pi ^0\)  electroproduction [13]}, [15]}, [16]}. The outcome of [18]} is
that the CLAS photoproduction data [17]} can be fitted with reasonable pion DAs and soft form
factors. Twist-3 effects may also be generated by twist-3 GPDs [26]} for which no
enhancement similar to \(\mu _\pi \)  parameter is known.
These twist-3 effects are expected to be small and are therefore neglected in [18]}
as well as in the present work.
